Character sums of Division polynomials twisted by multiplicative functions

Abstract

Let $E$ be an elliptic curve over the finite field ${\mathbb{F}}_p$, and $P\in E({\mathbb{F}}_p)$ be an ${\mathbb{F}}_p$-rational point. We obtain nontrivial estimates for multiplicative character sums associated with the division polynomials ${\psi }_n(P)$ twisted by several multiplicative functions, where ${\psi }_n(P)$ denotes the $n$-th division polynomial evaluated at $P$.

arXiv comment21 pages, incorporated some comments, more comments are welcome!